Internal Consistency In Models Of Optimal Resource Use Under Uncertainty

For several decades, economists have been concerned with the problem of optimal resource use under uncertainty. In many studies, researchers assume that prices evolve according to an exogenous stochastic process and solve the corresponding dynamic optimization problem to yield an optimal decision rule for exploitation of the resource. This study is motivated by our attempt to understand the relationship between efficiency in resource markets and optimal harvest decisions in which price is an exogenous state variable. The literature on optimal commodity storage finds that in a rational expectations equilibrium commodity prices are stationary and serially correlated. Yet recent papers on optimal timber harvesting that assume exogenous stationary prices generate harvest rules inconsistent with the price processes on which they are based. In this study, we investigate the appropriate form of the stochastic process governing prices of renewable resources. We develop a model in which timber is supplied by profit-maximizing managers with rational expectations and aggregate timber demand is subject to independent exogenous shocks. In contrast to earlier studies, prices are endogenously determined. Managers know the structure of the timber market and form expectations of future market equilibria in making optimal harvesting decisions. We show under general conditions that efficient timber prices are stationary and serially correlated. Stationarity and serial correlation are shown to arise from two sources: the occurrence of stock-outs (i.e., depletion of the inventory) and stock-dependent growth of the resource. Further, we show that prices retain these properties even in the absence of stock-outs. Simulations are used to further illustrate the analytical results. Our findings have implications for a large number of economic analyses of optimal resource use. First, our results reveal why extraction rules for renewable resources based on exogenous price specifications are internally inconsistent, even when the specification conforms to the stochastic behavior of prices generated by an efficient market. These prices arise in a particular structural environment, and if large numbers of resource managers adopt the harvesting rule, the underlying structural environment would change, and the price process would deviate from that used to derive the harvesting rule. Second, we show that there can be no gains from exploiting the stochasticity of resource prices in a rational expectations world, a finding that challenges the prescriptive policies for resource use found in many studies, including those on option values. Third, our results show that time-series analyses designed to test for the efficiency of renewable resource markets cannot distinguish prices generated in an efficient market from those generated in an inefficient market. Finally, we extend the literature on optimal storage. Previous models of commodity storage models are shown to be a special case of our model involving age-independent depreciation of the inventory.